The product rule

About this lesson

Calculus AB (AP-aligned) › Differentiation rules

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The product rule

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = x$ and $g(x) = x$ , so $f(x)g(x) = x^2$ . Is $(x^2)'$ equal to $f'(x)\,g'(x) = 1 \cdot 1$ ?

No
Yes

Key idea

Each factor takes its turn to be differentiated.

Product rule:

(f\,g)' = f'\,g + f\,g'

.

Worked example

$h(x) = x^3 e^x$ . Find $h'(x)$ .

f' = 3x^2

,

g' = e^x

f = x³, g = eˣ.

h'(x) = 3x^2 e^x + x^3 e^x

f'g + fg'.

h'(x) = x^2 e^x (3 + x)

Factor if you like.

Your turn, guided

Your turn: $h(x) = x\cos x$ .

f' = 1

,

g' = -\sin x

f = x, g = cos x.

h'(x) = \cos x - x\sin x

f'g + fg'.

Practice ★

$f(x) = x^{ 2 }\sin x$ . Find $f'(x)$ .
$f(x) = x^{ 3 }\sin x$ . Find $f'(x)$ .

$f(x) = (x + 3)e^x$ . Find $f'(x)$ .
$f(x) = (x + 8)e^x$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(2) = 4$ , $f'(2) = 4$ , $g(2) = -4$ and $g'(2) = -1$ . Find $(fg)'(2)$ .
$f(2) = -4$ , $f'(2) = -2$ , $g(2) = 2$ and $g'(2) = 3$ . Find $(fg)'(2)$ .
$f(2) = -1$ , $f'(2) = 1$ , $g(2) = 5$ and $g'(2) = -1$ . Find $(fg)'(2)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = x^{ 2 }\sin x$ . Find $f'(x)$ .
$f(x) = x^{ 3 }\sin x$ . Find $f'(x)$ .
$f(x) = x^{ 5 }\sin x$ . Find $f'(x)$ .

Summary card

Product rule

$(fg)' = f'g + fg'$ : differentiate one factor at a time, keep the other.

Watch out: $(fg)' \ne f'g'$ .

End of the lesson

Next lesson: The quotient rule Back to the chapter

Answers: The product rule

A

$x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}$
$x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}$

B

$\left(x + 4\right) e^{x}$
$\left(x + 9\right) e^{x}$

C

$-20$
$-16$
$6$

D

$x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}$
$x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}$
$x^{5} \cos{\left(x \right)} + 5 x^{4} \sin{\left(x \right)}$

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